Metric and imperial are exactly related
Since 1959 the imperial units have been defined in terms of the metre, so conversions between the two systems are exact rather than approximate. One inch is exactly 25.4 millimetres. One yard is exactly 0.9144 metres. Everything else follows from those two definitions.
This matters more than it sounds. Older conversion tables built on the pre-1959 US survey foot differ by about two parts per million — negligible for a room, but a measurable error across a construction site or a land survey, which is why the US survey foot was formally retired at the end of 2022.
Where the mistakes actually happen
Unit slips are expensive. The Mars Climate Orbiter was lost in 1999 because one team supplied thrust data in pound-seconds while the other assumed newton-seconds. Closer to home, the usual errors are simpler: confusing a nautical mile (1852 m, used in aviation and shipping) with a statute mile (1609.344 m), or reading a height like 5'10" as 5.10 feet rather than 5.833 feet.
For heights given in feet and inches, convert the inches to a fraction of a foot first — 10 inches is 10/12, or 0.8333 feet — then convert the total. 5'10" is 5.8333 feet, which is 1.778 metres.